Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

doi:10.1007/s42967-024-00420-y

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 149-176.doi: 10.1007/s42967-024-00420-y

Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

Wei Zheng, Yan Xu

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

收稿日期:2023-10-24 修回日期:2024-03-24 出版日期:2026-02-20 发布日期:2026-02-11
通讯作者: Yan Xu,E-mail:yxu@ustc.edu.cn E-mail:yxu@ustc.edu.cn
作者简介:Wei Zheng,E-mail:zhengw18@mail.ustc.edu.cn
基金资助:
Research of Yan Xu was supported by the NSFC under grant No. 12071455.

Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

Wei Zheng, Yan Xu

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received:2023-10-24 Revised:2024-03-24 Online:2026-02-20 Published:2026-02-11
Contact: Yan Xu,E-mail:yxu@ustc.edu.cn E-mail:yxu@ustc.edu.cn
Supported by:
Research of Yan Xu was supported by the NSFC under grant No. 12071455.

摘要/Abstract

摘要： This paper presents a novel approach to preserving invariants in the time-implicit numerical discretization of high-order nonlinear wave equations. Many high-order nonlinear wave equations have an infinite number of conserved quantities. Designing time-implicit numerical schemes that preserve many conserved quantities simultaneously is challenging. The proposed method utilizes Lagrange multipliers to reformulate the local discontinuous Galerkin (LDG) discretization as a conservative discretization. Combining an implicit spectral deferred correction method for time discretization can achieve a high-order scheme in both temporal and spatial dimensions. We use the Korteweg-de Vries equations as an example to illustrate the implementation of the algorithm. The algorithm can be easily generalized to other nonlinear wave problems with conserved quantities. Numerical examples for various high-order nonlinear wave equations demonstrate the effectiveness of the proposed methods in preserving invariants while maintaining the high accuracy.

关键词: Local discontinuous Galerkin (LDG) method, Conservative and dissipative schemes, Korteweg-de Vries type equations, Spectral deferred correction (SDC) method, Lagrange multiplier

Abstract: This paper presents a novel approach to preserving invariants in the time-implicit numerical discretization of high-order nonlinear wave equations. Many high-order nonlinear wave equations have an infinite number of conserved quantities. Designing time-implicit numerical schemes that preserve many conserved quantities simultaneously is challenging. The proposed method utilizes Lagrange multipliers to reformulate the local discontinuous Galerkin (LDG) discretization as a conservative discretization. Combining an implicit spectral deferred correction method for time discretization can achieve a high-order scheme in both temporal and spatial dimensions. We use the Korteweg-de Vries equations as an example to illustrate the implementation of the algorithm. The algorithm can be easily generalized to other nonlinear wave problems with conserved quantities. Numerical examples for various high-order nonlinear wave equations demonstrate the effectiveness of the proposed methods in preserving invariants while maintaining the high accuracy.

Key words: Local discontinuous Galerkin (LDG) method, Conservative and dissipative schemes, Korteweg-de Vries type equations, Spectral deferred correction (SDC) method, Lagrange multiplier

中图分类号:

Wei Zheng, Yan Xu. Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 149-176.

参考文献

[1] Bihlo, A., Valiquette, F.: Symmetry-preserving finite element schemes: an introductory investigation. SIAM J. Sci. Comput. 41(5), A3300-A3325 (2019)
[2] Chen, Y., Dong, B., Pereira, R.: A new conservative discontinuous Galerkin method via implicit penalization for the generalized Korteweg-de Vries equation. SIAM J. Numer. Anal. 60(6), 3078-3098 (2022)
[3] Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545-581 (1990)
[4] Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90-113 (1989)
[5] Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411-435 (1989)
[6] Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440-2463 (1998)
[7] Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys 141(2), 199-224 (1998)
[8] Cui, Y., Mao, D.: Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation. J. Comput. Phys. 227(1), 376-399 (2007)
[9] Deuflhard, P.: Newton methods for nonlinear problems: affine invariance and adaptive algorithms. In: Springer Series in Computational Mathematics, vol. 35. Springer, Heidelberg (2011)
[10] Dey, B., Khare, A., Nagaraja Kumar, C.: Stationary solitons of the fifth order KdV-type. Equations and their stabilization. Phys. Lett. A 223(6), 449-452 (1996)
[11] Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241-266 (2000)
[12] Guo, R., Xia, Y., Xu, Y.: Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations. J. Comput. Phys. 338, 269-284 (2017)
[13] Guo, R., Xu, Y.: Fast solver for the local discontinuous Galerkin discretization of the KdV type equations. Commun. Comput. Phys. 17(2), 424-457 (2015)
[14] Guo, R., Xu, Y.: Semi-implicit spectral deferred correction method based on the invariant energy quadratization approach for phase field problems. Commun. Comput. Phys. 26(1), 87-113 (2019)
[15] Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Berlin (2006)
[16] Karaduman, G., Yang, M., Li, R.-C.: A least squares approach for saddle point problems. Jpn. J. Ind. Appl. Math. 40(1), 95-107 (2023)
[17] Karpman, V.I.: Stabilization of soliton instabilities by higher order dispersion: KdV-type equations. Phys. Lett. A 210(1), 77-84 (1996)
[18] Kim, P.: Invariantization of numerical schemes using moving frames. BIT Numer. Math. 47(3), 525-546 (2007)
[19] Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. (5) 39(240), 422-443 (1895)
[20] Kuznetsov, E.A., Rubenchik, A.M., Zakharov, V.E.: Soliton stability in plasmas and hydrodynamics. Phys. Rep. 142(3), 103-165 (1986)
[21] Levy, D., Shu, C.-W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196(2), 751-772 (2004)
[22] Li, D., Li, X., Zhang, Z.: Linearly implicit and high-order energy-preserving relaxation schemes for highly oscillatory Hamiltonian systems. J. Comput. Phys. 477, 111925 (2023)
[23] Liu, H., Yi, N.: A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation. J. Comput. Phys. 321, 776-796 (2016)
[24] McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 1021-1045 (1999)
[25] Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9, 1204-1209 (1968)
[26] Montoison, A., Orban, D.: GPMR: an iterative method for unsymmetric partitioned linear systems. SIAM J. Matrix Anal. Appl. 44(1), 293-311 (2023)
[27] Nouri, F.Z., Sloan, D.M.: A comparison of Fourier pseudospectral methods for the solution of the Korteweg-de Vries equation. J. Comput. Phys. 83(2), 324-344 (1989)
[28] Nowak, U., Weimann, L.: A family of Newton codes for systems of highly nonlinear equations. Technical Report TR-91-10, ZIB, Takustr. 7, 14195 Berlin (1992)
[29] Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41(4), 045206 (2008)
[30] Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report, Los Alamos Scientific Lab., N. Mex., USA (1973)
[31] Rosenau, P., Hyman, J.M.: Compactons: solitons with finite wavelength. Phys. Rev. Lett. 70, 564-567 (1993)
[32] Simo, J.C., Tarnow, N., Wong, K.K.: Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Methods Appl. Mech. Eng. 100(1), 63-116 (1992)
[33] van der Vegt, J.J.W., Xia, Y., Xu, Y.: Positivity preserving limiters for time-implicit higher order accurate discontinuous Galerkin discretizations. SIAM J. Sci. Comput. 41(3), A2037-A2063 (2019)
[34] Xia, Y., Xu, Y., Shu, C.-W.: Efficient time discretization for local discontinuous Galerkin methods. Discrete Contin. Dyn. Syst. Ser. B 8(3), 677-693 (2007)
[35] Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for three classes of nonlinear wave equations. J. Comput. Math. 22, 250-274 (2004)
[36] Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196(37/38/39/40), 3805-3822 (2007)
[37] Yin, X., Cao, W.: A class of efficient Hamiltonian conservative spectral methods for Korteweg-de Vries equations. J. Sci. Comput. 94(1), 10-29 (2023)
[38] Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240-243 (1965)
[39] Zhang, Q., Xia, Y.: Conservative and dissipative local discontinuous Galerkin methods for Korteweg-de Vries type equations. Commun. Comput. Phys. 25(2), 532-563 (2019)
[40] Zhao, P.F., Qin, M.Z.: Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation. J. Phys. A 33(18), 3613-3626 (2000)

Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

编辑推荐

Metrics

本文评价

[1]	Qiaoqiao Dai, Dongxia Li. The L2-1_σ/LDG Method for the Caputo Diffusion Equation with a Variable Coefficient[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1378-1397.
[2]	Zhen Wang. L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 203-227.
[3]	Meiqi Tan, Juan Cheng, Chi-Wang Shu. The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 115-150.